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Hindawi Publishing Corporation
Boundary Value Problems
Volume 2011, Article ID 214289, 19 pages
doi:10.1155/2011/214289
Research Article
Multiple Solutions of p-Laplacian with
Neumann and Robin Boundary Conditions for
Both Resonance and Oscillation Problem
Jing Zhang and Xiaoping Xue
Department of Mathematics, Harbin Institute of Technology, Harbin 150025, China
Correspondence should be addressed to Jing Zhang,
Received 29 June 2010; Revised 7 November 2010; Accepted 18 January 2011
Academic Editor: Sandro Salsa
Copyright q 2011 J. Zhang and X. Xue. This is an open access article distributed u nder the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We discuss Neumann and Robin problems driven by the p-Laplacian with jumping nonlinearities.
Using sub-sup solution method, Fuc
´
ık spectrum, mountain pass theorem, degree theorem together
with suitable truncation techniques, we show that the Neumann problem has infinitely many
nonconstant solutions and the Robin problem has at least four nontrivial solutions. Furthermore,
we study oscillating equations with Robin boundary and obtain infinitely many nontrivial
solutions.
1. Introduction
Let Ω be a bounded domain of R
n
with smooth boundary ∂Ω, we consider the following
problems:
i Neumann problem:
−Δ
p
u α
|
u
|
p−2
u f
x, u
, in Ω,
∂u
∂ν
0, on ∂Ω,
p
1
ii Robin problem:
−Δ
p
u α
|
u
|
p−2
u f
x, u
, in Ω,
|
∇u
|
p−2
∂u
∂ν
b
x
|
u
|
p−2
u 0, on ∂Ω,
p
2
2 Boundary Value Problems
where Δ
p
u div|∇u|
p−2
∇u is the p-Laplacian operator of u with 1 <p<∞, α>0,
bx ∈ L
∞
∂Ω, bx ≥ 0, and bx
/
0on∂Ω, fx, 00fora.e.x ∈ Ω,and∂u/∂ν
denotes the outer normal derivative of u with respect to ∂Ω.Ourpurposeistoshow
the multiplicity of solutions to p
1
and p
2
.
It is known that p
1
and p
2
are the Euler-Lagrange equations of the functionals
J
1
u
1
p
Ω
|
∇u
|
p
dx
α
p
Ω
|
u
|
p
dx −
Ω
F
x, u
dx,
J
2
u
1
p
Ω
|
∇u
|
p
dx
α
p
Ω
|
u
|
p
dx
1
p
∂Ω
b
x
|
u
|
p
ds −
Ω
F
x, u
dx,
1.1
respectively, defined on the Sobolev space W
1,p
Ω,whereFx, u
u
0
fx, sds. The critical
points of functionals correspond to the weak solutions of problems. In Li 1 and Zhang et al.
2, the authors study the existence and multiple solutions of p
1
and p
2
using the critical
points theory for the semilinear case p 2. There also have been some papers dealing with the
quasilinear case p
/
2 using the critical point theory, and some existence results of solutions
have been generalized to this case in the work of Perera 3, Zhang et al. 4,andZhang-Li5.
Most of these papers use the minimax arguments, and nontrivial solutions are obtained with
the assumption that the nonlinearity is superlinear at 0. In this paper, we give the nontrivial
solutions of p
1
and p
2
with a jumping nonlinearity when the asymptotic limits of the
nonlinearity fall in the regions formed by the curves of the Fuc
´
ık spectrum. Our technique is
based on mountain pass theorem, computing the critical groups and Fuc
´
ıkspectrum.
Our general assumptions are the following.
f
1
There is constant C>0suchthatfx, t satisfies the following subcritical condi-
tions:
f
x, t
≤ C
|
t
|
q
1
for every x ∈ Ω,t∈ R, 1.2
with p − 1 <q<p
∗
− 1, where p
∗
np/n − p if n>p,andp
∗
∞ if n 1, 2, ,p.
f
2
∃ sequence {a
i
} and {b
i
},wherea
i
,b
i
∈ R, i 1, 2, , which satisfy a
i
> 0, b
i
< 0
and a
i
∞, b
i
−∞as i →∞. And at the same time {a
i
}, {b
i
} satisfy
f
x, a
i
αa
p−1
i
,f
x, b
i
−α
|
b
i
|
p−1
, for every x ∈ Ω
1.3
which means that {a
i
}, {b
i
} are constant solution sequences of p
1
.
Let a
0
b
0
0, fx, t <αt
p−1
if t ∈ a
i
,a
i1
,wherei is an odd number, i ≥ 1;
fx, t >αt
p−1
if t ∈ a
i
,a
i1
,wherei is an even number, i ≥ 0; fx, t < −α|t|
p−1
if t ∈ b
i1
,b
i
,
where i is an even number, i ≥ 0; fx, t > −α|t|
p−1
if t ∈ b
i1
,b
i
,wherei is an odd number,
i ≥ 1, for every x ∈ Ω.
f
3
For all t
/
a
i
,b
i
, f is C
1
; f
−
x, a
i
/
f
x, a
i
, f
−
x, b
i
/
f
x, b
i
,wherei is an even
number, i ≥ 2, f
−
x, t, f
x, t denote the left and the right derivatives of f at t,
respectively.
Boundary Value Problems 3
f
4
Let a, bf
x, a
i
− α, f
−
x, a
i
− α for i is an even number, i ≥ 2. For a, b ∈ R
2
,
the problem
−Δ
p
u a
u − c
p−1
− b
u − c
−
p−1
, in Ω,
∂u
∂ν
0, on ∂Ω,
1.4
only has constant solution c,whereu−c
±
xmax{±u−c, 0} and c is a constant.
And f
i−
x, a
i
− α>λ
2
, f
i
x, a
i
− α>λ
2
for i is an even number, i ≥ 2, where
f
i
x, t
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
0,t<0,
f
x, t
, 0 ≤ t ≤ a
i
,
f
x, a
i
,t>a
i
,
1.5
and f
i−
x, a
i
,f
i
x, a
i
denote the left and the right derivatives of f
i
at a
i
, respectively, and
λ
2
is the second of the eigenvalue problems with Neumann boundary value condition.
f
5
∃m>α,suchthatfx, tm|t|
p−2
t is increasing in t.
In particular, from f
2
, we know that p
1
has infinitely many constant solutions, a.e.,
{a
i
}, {b
i
}, i 0, 1, 2, In this paper, we mainly discuss whether it has many nonconstant
solutions and what their locations are.
Then we have the main results of this paper.
Theorem 1.1. Assume that (f
1
)–(f
5
)hold.Thenp
1
has infinitely many nonconstant solutions.
Moreover, if one chooses some order intervals which have two pairs of strict constant sub-sup solutions,
then p
1
has at least two nonconstant solutions in some order intervals.
Furthermore, if we assume that f
−
x, 0
/
f
x, 0 under the same c onditions as in
Theorem 1.1, we can have at least one sign-changing solution which is of mountain pass type
from the mountain pass theorem in order interval. When we discuss multiple solutions of
p
1
, we notice that there may be infinitely many sign-changing s olutions under stronger
assumptions. In fact, if we give more assumptions,we can obtain infinitely many sign-
changing solutions.
We assume the following.
F Fx, t > λ
2
α ε
0
/pt
p
, |t|≥M, M is large enough, where λ
2
is the second
eigenvalue of Neumann problem of −Δ
p
and ε
0
> 0.
Corollary 1.2. Under the same conditions as in Theorem 1.1,(F)andf
−
x, 0
/
f
x, 0,thenone
can get infinitely many sign- changing solutions for p
1
which are of mountain pass type or not
mountain pass type but with positive local degree.
For the Robin problem, if ∃M
1
> 0, M
2
> 0suchthatfx, M
1
0, fx, −M
2
0for
a.e. x ∈ Ω, then we give the following assumptions:
4 Boundary Value Problems
g
1
f ∈ C
1
Ω × R
1
\{0}, f
−
x, 0
/
f
x, 0,andmin{f
x, 0,f
−
x, 0} >λ
1
α for a.e.
x ∈ Ω,wheref
−
x, 0, f
x, 0 denote the left and the right derivatives of f at 0,
respectively, and λ
1
is the first eigenvalue of Robin problem of −Δ
p
;
g
2
let a, bf
x, 0 − α, f
−
x, 0 − α.Fora, b ∈ R
2
,theproblem
−Δ
p
u a
u
p−1
− b
u
−
p−1
, in Ω,
|
∇u
|
p−2
∂u
∂ν
b
x
|
u
|
p−2
u 0, on ∂Ω,
1.6
only has trivial solution 0, where u
±
xmax{±u, 0}.
In this case, we have the following.
Theorem 1.3. Assume that (f
1
), (f
5
), (g
1
), (g
2
) hold. Then one has at least four nontrivial solutions
of problem p
2
.
Furthermore, we give the following stronger assumption:
F
Fx, t > λ
2
α ε
0
/p
Ct
p
, |t|≥M, Fx, u
u
0
fx, sds, u ∈ E
2
,whereE
2
{u ∈ W
1,p
Ω : u kϕ
1
tϕ
2
},
C C
2
/2bx
L
∞
∂Ω
.HereC is the imbedding
constant of Sobolev Trace Theorem see 6, M is large enough, ε
0
is small enough,
λ
2
is the second of the eigenvalue problems with Robin boundary value condition,
and ϕ
1
, ϕ
2
are the first and the second eigenfunction, respectively.
Then we have the following.
Corollary 1.4. Assume that f is satisfied as in Theorem 1.3 and (F
), then one can have infinitely
many sign-changing solutions for p
2
which are of mountain pass type or not mountain pass type but
with positive local degree.
In the oscillating problems of Robin boundary, a.e., f
2
holds. We make the following
assumption.
F
Ω
Fx, tϕ
1
dx ≥ λ
1
α ε
0
/p
Ct
p
Ω
ϕ
p
1
dx, |t|≥M,whereϕ
1
is the first
eigenvalue of the Robin problem and
Ω
ϕ
p
1
dx 1.
Then we have the following.
Theorem 1.5. Assume that f is satisfied as in Theorem 1.3 and (f
2
), (F
), one can get infinitely
many nontrivial solutions of problem p
2
. Some of them are minimum points; others are mountain
pass points.
2. Preliminaries
Now we recall the notion of critical groups of an isolated critical point u of a C
1
functional J
briefly. Let U ⊂ M be an isolated neighborhood of u such that there are no critical points of J
in U \{u}; M is a Banach space. The critical groups of u are defined as
C
q
J, u
H
q
J
c
∩ U,
J
c
\
{
u
}
∩ U; G
,q 0, 1, 2, , 2.1
Boundary Value Problems 5
where c Ju and J
c
{u ∈ M|Ju ≤ c} is a level set of J and H
q
X, Y; G are singular
relative homology groups with a Abelian coefficient group G, Y ⊂ X, q 0, 1, 2, Theyare
independent of the choices of U, hence are well defined. Use H
q
X; G to stand for the qth
singular cohomology group with an Abelian coefficient group G; from now on we denote it
by H
q
X. Assume that J ∈ C
2
M, R, and a critical point u is called nondegenerate if the
Hessian J
u at this point has a bounded inverse. Let u be a nondegenerate critical point of
J; we call the dimension of the negative space corresponding to the spectral decomposition
of J
u, that i s, the dimension of the subspace of negative eigenvectors of J
u,theMorse
index of u, and denote it by indJ
u.IfC
1
J, u
/
0, then we call an isolated critical point u
of J as a mountain pass point. For the details, we refer to 7.
We have the following basic facts on the critical groups for an isolated critical point
of J.
a Let u be is an isolated minimum point of J,thenC
q
J, uδ
q0
G.
b If J ∈ C
2
M, R and u is a nondegenerate critical point of J with Morse index j,then
C
q
J, uδ
qj
G.
Definition 2.1. If any sequence {u
k
}⊂M which satisfies Ju
k
→ c and J
u
k
→ 0 k →∞
has a convergent subsequence, one says that J satisfies the PS
c
condition. If J satisfies PS
c
condition for all c ∈ R,onesaysthatJ satisfies the PS condition.
Lemma 2.2 see 8. Assume that u
and u are, respectively, lower and upper solutions for the
problem
−Δ
p
u g
x, u
, in Ω,
|
∇u
|
p−2
∂u
∂ν
b
x
|
u
|
p−2
u 0, on ∂Ω,
2.2
with u
≤ u a.e. in Ω,wheregx, s is a Carath
´
eodory function on Ω × R with the property that, for
any s
0
> 0, there exists a constant A such that |gx, s|≤A for a.e. x ∈ Ω and all s ∈ −s
0
,s
0
.
Consider the associated functional
Φ
u
:
1
p
Ω
|
∇u
|
p
−
Ω
G
x, u
,
2.3
where Gx, u :
s
0
gx, tdt and the interval M : {u ∈ W
1,P
Ω : u ≤ u ≤ u a.e. in Ω}. Then the
infimum of Φ on M is achieved at some u,andsuchau is a solution of the above problem.
In what follows, we set X W
1,p
Ω which is is uniformly convex 1 <p<∞ and
equipped with the norm u
Ω
|∇u|
p
dxmα
Ω
|u|
p
dx
1/p
.LetE be a Hilbert space and
P
E
⊂ E a closed convex cone such that X is densely embedded in E. Assume that P X ∩ P
E
,
P has nonempty interior
˙
P and any order interval is bounded. It is well known that PS
condition implies the compactness of the critical set at each level c ∈ R,onthecaseofthe
above condition. Then we assume the following:
J
1
J ∈ C
2
E, R and satisfies PS condition in E and deformation property in X;
6 Boundary Value Problems
J
2
∇J id − K
E
,whereK
E
: E → E is compact. K
E
X ⊂ X and the restriction
K K
E
|
X
: X → X is of class C
1
and strongly preserving, that is, u v ⇔ u−v ∈
˙
P;
J
3
J is bounded from below on any order interval in X.
Lemma 2.3 Mountain pass theorem in half-order intervals, sup-solutions case see 9.
Suppose that J satisfies (J
1
)–(J
3
). v
1
<v
2
is a pair of strict supersolution of ∇J 0. v
0
<v
1
is
a subsolution of ∇J 0. Suppose that v
0
,v
1
and v
0
,v
2
are admissible invariant sets for J.IfJ
has a local strict minimizer w in v
0
,v
2
\ v
0
,v
1
.ThenJ has mountain pass points u
0
in v
0
,v
2
\
v
0
,v
1
.
Lemma 2.4 Mountain pass theorem in order intervals see 10. Suppose that J satisfies (J
1
)–
(J
3
)and{v
1
,v
2
},{ω
1
,ω
2
} are two pairs of strict sub-sup solutions of ∇J 0 in X with v
1
<ω
2
,
v
1
,v
2
∩ ω
1
,ω
2
∅.ThenJ has a mountain pass point u
0
,u
0
∈ v
1
,ω
2
\ v
1
,v
2
∪ ω
1
,ω
2
.
More precisely, let v
0
be the maximal minimizer of J in v
1
,v
2
and ω
0
the minimal m inimizer of J in
ω
1
,ω
2
.Thenv
0
u
0
ω
0
.Moreover,C
1
J, u
0
, the critical group of J at u
0
, is nontrivial.
Remark 2.5. a Lemma 2.4 still holds if J ∈ C
1
E, R, K is of class C
0
see 10.
b For X W
1,p
Ω,wedefineg
p
t : |t|
p−2
t. From assumption f
5
,thereexists
m>αsuch that fx, u − α|u|
p−2
u mg
p
u is strictly increasing in u. The assumption is not
essential but is assumed for simplicity. If such m does not exist then we can approximate f
by a sequence of functions so that m as above exists, and obtain the solutions by passing to
limits. For m>α, we need the operator
A
m
: X −→ X, A
m
u
−Δ
p
mg
p
·
−1
x, u
mg
p
u
.
2.4
From 11, we know that A
m
is compact, that is, it is continuous and maps bounded subsets
of X into relatively compact subsets of X.Since−Δ
p
u mg
p
u is a positive operator,
K :
−Δ
p
u mg
p
u
−1
f
x, u
− α
|
u
|
p−2
u mg
p
u
2.5
is strongly orderpreserving. Fr om the above discussion, we have the mountain pass theorem
in order intervals of J
1
and J
2
.
Next, let us recall some notions and known results on Fuc
´
ıkspectrum.
The Fuc
´
ıkspectrumofp-Laplacian on W
1,p
Ω is defined as the set Σ
p
of those points
a, b ∈ R
2
for which the problem
−Δ
p
u a
u
p−1
− b
u
−
p−1
,u∈ W
1,p
Ω
2.6
has nontrivial solutions. Here u
±
xmax{±u, 0}.
Boundary Value Problems 7
For the semilinear case p 2, it is known that Σ
2
consists, at least locally, of curves
emanating from the points λ
l
,λ
l
where {λ
l
}
l∈N
are the distinct eigenvalues of −Δsee, e.g.,
12. It was shown in Schechter 13 that Σ
2
contains continuous and strictly decreasing
curves C
l
1
, C
l
2
through λ
l
,λ
l
such that the points in the square Q
l
λ
l−1
,λ
l1
2
that are
either below the lower curve C
l
1
or above the upper curve C
l
2
are free of Σ
2
, while the points
on the curves are in Σ
2
when they do not coincide. The points in the region between the
curves may or may not belong to Σ
2
.
AsshowninLindqvist14 that the first eigenvalue λ
1
of −Δ
p
is positive, simple and
admits a strictly positive eigenfunction ϕ
1
,soΣ
p
contains the two lines λ
1
× R and R × λ
1
.
This generalized notion of spectrum was introduced for the semilinear case p 2 in the 1970s
by Fuc
´
ık 12 in connection with jumping nonlinearities. A first nontrivial curve C
2
in Σ
p
through λ
2
,λ
2
that is continuous, strictly decreasing, and asymptotic to λ
1
× R and R × λ
1
at infinity was constructed and variationally characterized by a mountain-pass procedure in
Cuesta et al. 15.
Consider the problem
−Δ
p
u a
u − c
p−1
− b
u − c
−
p−1
, in Ω,
∂u
∂ν
0, on ∂Ω,
2.7
−Δ
p
u a
u
p−1
− b
u
−
p−1
, in Ω,
|
∇u
|
p−2
∂u
∂ν
b
x
|
u
|
p−2
u 0, on ∂Ω,
2.8
from the variational point of view; solutions of 2.7 and 2.8 are the critical points of the
functional
I
1
u
I
1
u, a, b
Ω
|
∇u
|
p
− a
u − c
p
− b
u − c
−
p
dx,
I
2
u
I
2
u, a, b
Ω
|
∇u
|
p
− a
u
p
− b
u
−
p
dx
1
p
∂Ω
b
x
|
u
|
p
ds,
2.9
respectively, where c is a constant.
If a, b does not belong to Σ
p
, c is the constant solution of 2.7,thatis,c is an isolated
critical point of I
1
; 0 is the trivial solution of 2.8, that is, 0 is an isolated critical point of
I
2
, then from the definition of critical group, we have the C
q
I
1
,c and C
q
I
2
, 0 defined, q
0, 1, 2 , Now, we give some results relative to the computation of the critical groups which
are the results of Dancer and Perera 16.LetC
11
−∞,λ
1
× λ
1
∪ λ
1
× −∞,λ
1
and
C
12
λ
1
× λ
1
, ∞ ∪ λ
1
, ∞ × λ
1
.
8 Boundary Value Problems
Lemma 2.6. i If a, b lies below C
11
,thenC
q
I, cδ
q0
Z.
ii If a, b lies between C
11
and C
12
,thenC
q
I, c0 for all q.
iii If a, b lies between C
12
and C
2
,thenC
q
I, cδ
q1
Z.
iv If a, b does not belong to Σ
p
, but lies above C
2
,thenC
q
I, c0 for q 0, 1.
Denote
I
s
u
Ω
|
∇u
|
p
− s
u − c
p
,u∈ X,
2.10
and
I
s
is the restriction of I
s
to the C
1
manifold
S
u ∈ X :
Ω
|
u − c
|
p
1
.
2.11
As noted in 16, the critical groups of I are related to the homology gro ups of sublevel sets
of
I
a−b
.Wehavethat
I|
S
I
a−b
− b,
2.12
so the sublevel sets
I
d
{
u ∈ X : I
u
≤ d
}
,
I
d
s
u ∈ S :
I
s
≤ d
2.13
are related by
I
d
∩ S
I
db
a−b
.
2.14
Lemma 2.7. If a, b does not belong to Σ
p
,then
C
q
I, c
∼
⎧
⎨
⎩
δ
q0
Z, if
I
b
a−b
∅,
H
q−1
I
b
a−b
, otherwise,
2.15
where
H
q
denote reduced homology groups. It also holds with
I
b
a−b
replaced by
b
{u ∈ S :
I
a−b
u >b}.
Boundary Value Problems 9
3. The Proof of the Main Results
Let
f
i
x, t
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
0,t<0,
f
x, t
, 0 ≤ t ≤ a
i
,
f
x, a
i
,t>a
i
.
F
i
x, t
t
0
f
i
x, s
ds
J
1i
u
1
p
Ω
|
∇u
|
p
dx
α
p
Ω
|
u
|
p
dx −
Ω
F
i
x, u
dx.
3.1
It is well known that critical points of J
1i
correspond to weak solutions of the following
equation:
−Δ
p
u α
|
u
|
p−2
u f
i
x, u
, in Ω,
∂u
∂ν
0, on ∂Ω.
p
We have that f
i
x, t ∈ C
0
R, R and J
1i
∈ C
1
E, R. We can discuss similar case for b
i
.
Next, we give the relation of the solutions of p
and the solutions of p
1
,thatis,
Lemma 3.2 below. In order to prove Lemma 3.2, we firstly give the comparison principle.
Let
L
p
: −Δ
p
a
x
|
u
|
p−2
u,
λ
1,p
a
inf
Ω
|
∇u
|
p
a
x
|
u
|
p
dx, u ∈ W
1,p
0
Ω
,
Ω
|
u
|
p
dx 1
.
3.2
Lemma 3.1 comparison principle see 17. Assume a ∈ L
∞
Ω, λ
1,p
a > 0.TheL
p
u ∈
L
∞
Ω with u|
∂Ω
∈ C
1α
∂Ω,andL
p
u ≤ 0 with u ∈ W
1,p
Ω ∩ L
∞
Ω,thenu ≤ 0.
Lemma 3.2. If u
i
x is a solution of p
,thenu
i
x is also a solution of p
1
and satisfies 0 ≤
u
i
x ≤ a
i
, i 1, 2,
Proof. Suppose that the conclusion is false. Now, consider the domain U
i
{x ∈ Ω | u
i
x >
a
i
},thenwehave
−Δ
p
u f
i
x, u
− α
|
u
|
p−2
u ≤ 0, in U
i
,
u a
i
, on ∂U
i
,
3.3
where −Δ
p
u f
i
x, u − α|u|
p−2
u fx, a
i
− α|u|
p−2
u ≤ fx, a
i
− αa
p−1
i
0 ,x ∈ U
i
by the
definition of f
i
x, u. By the comparison principle, we can conclude that u
i
x ≤ 0inU
i
.Itis
a contradiction, so we have that U
i
∅,thatis,u
i
x ≤ a
i
.
Similarly, we consider V
i
{x ∈ Ω | u
i
x < 0}, by the comparison principle; we also
get the contradiction, so we have that V
i
∅,thatis,u
i
x ≥ 0. From the above discussion, we
10 Boundary Value Problems
have that 0 ≤ u
i
x ≤ a
i
, i 1, 2, and f
i
x, ufx, u
i
,sou
i
x is a solution of p
1
.This
completes the proof of the lemma.
Remark 3.3. From the above discussion, by applying Lemma 3.2, we know that solutions of
p
are also the solutions of p
1
if we want to pr ove Theorem 1.1, we only need to prove that
p
has infinitely many nonconstant solutions under the assumptions as in Theorem 1.1 and
p
has two nonconstant solutions in every order interval.
Theorem 3.4. There ar e infinitely many nonconstant solutions of p
. Moreover, if there exists some
order intervals which have two pairs of strict constant sub-sup solutions, then there are at least two
nonconstant solutions in these order intervals.
Proof. We treat the case of a
i
; the other case of b
i
is proved by a similar argument.
If f
2
holds, then
−Δ
p
a
i
0 f
i
x, a
i
− αa
p−1
i
, for a.e.x∈ Ω,
3.4
so {a
i
} are all positive constant solutions of p
. Assuming that i is large enough and i is an
even number, we also infer that {a
2k−1
} are local minimums, k 1, 2, ,i/2. So we get u
2k−1
and u
2k−1
a strict subsolution and sup-solution pair for p
, satisfying u
2k−1
<a
2k−1
< u
2k−1
for each k, k 1, 2, ,i/2.
Now, we study the order interval u
1
, u
3
in X which includes two suborder intervals
u
1
, u
1
and u
3
, u
3
, a
2
∈ u
1
, u
3
.
We infer that J
1i
u satisfies deformation properties and is bounded from below on
u
1
, u
3
and so we get a mountain pass point u
1
∈ u
1
, u
3
\ u
1
, u
1
∪ u
3
, u
3
according to
mountain pass theorem in order interval, we have that C
1
J
1i
,u
1
is nontrivial.
From assumption f
3
, we know that the left and the right derivatives of f
i
at a
2
are
different; we consider the problem
−Δ
p
u f
i
x, u
− α
|
u
|
p−2
u, in Ω,
∂u
∂ν
0, on ∂Ω,
3.5
where f
i
∈ CΩ × R and as u → a
2
we have
f
i
x, u
− α
|
u
|
p−2
u
f
i
x, a
2
− α
u − a
2
p−1
−
f
i−
x, a
2
− α
u − a
2
−
p−1
◦
|
u − a
2
|
p−1
.
3.6
We take a f
i
x, a
2
− α,b f
i−
x, a
2
− α, then from assumption f
4
and the definition of
Σ
p
, we know that a, b does not belong to Σ
p
. So, we have the following.
1 If a, b does not belong to Σ
p
, but lies above C
2
,then
C
q
J
1i
,a
2
0forq 0, 1 3.7
Boundary Value Problems 11
by Lemma 2.6iv.Inthiscase,C
1
J
1i
,a
2
0, so C
q
J
1i
,a
2
C
q
J
1i
,u
1
,andwe
have u
1
/
a
2
.
2 Denote
J
a−b
u
Ω
|
∇u
|
p
−
a − b
u − a
2
p
,u∈ X,
3.8
and
J
a−b
is the restriction of J
a−b
to the C
1
manifold
S
u ∈ X :
Ω
|
u − a
2
|
p
1
,
3.9
where a f
i
x, a
2
− α,b f
i−
x, a
2
− α as shown above.
From f
4
, we know that a, b does not belong to Σ
p
,andif
J
a−b
u >b,a.e.
J
b
a−b
∅,
then
C
q
J
1i
,a
2
δ
q0
Z 3.10
by Lemma 2.7.Inthiscase,C
1
J
1i
,a
2
0, so C
q
J
1i
,a
2
C
q
J
1i
,u
1
, and we have u
1
/
a
2
.
Similarly, applying the mountain pass theorem in order interval to u
3
, u
5
which
contain two sub-order intervals u
3
, u
3
and u
5
, u
5
, we get a mountain pass point u
2
and
prove that C
q
J
1i
,a
4
C
q
J
1i
,u
2
,sou
2
/
a
4
from Lemmas 2.6 and 2.7.
We let the procedure go on. So i/2 − 1 mountain pass points are available which are
nonconstant solutions of p
,wherei is large enough and i is an even number. Then we have
infinitely many nonconstant positive solutions of p
by the arbitrary of i.
We can discuss the similar case for b
i
and get infinitely many nonconstant negative
solutions.
Now, we discuss the solutions in u
1
, u
3
more deeply. Since u
1
is a mountain pass
point, for the Leray-Schauder degree of id − K
i
, we have the computing formular
deg
id − K
i
,B
u
1
,r
, 0
−1, 3.11
where r>0 is small enough, K
i
−Δ
p
m αg
p
·
−1
f
∗
i
|
X
: X → X is of class C
0
and strongly preserving, f
∗
i
x, uf
i
x, umg
p
usee Remark 2.5b. Then according to
Poincar
´
e-Hopf formular for C
1
case and the computation of C
q
J
1i
,a
2
,wehave
index
J
1i
,a
2
−1
l
.
3.12
Furthermore, for minimum points a
1
, a
3
,
C
q
J
1i
,a
1
∼
δ
q0
G, C
q
J
1i
,a
3
∼
δ
q0
G. 3.13
12 Boundary Value Problems
From the additivity of Leray-Schauder degree and Theorem 1.1 in 10,wecanget
1 deg
id − K
i
,
u
1
, u
3
, 0
deg
id − K
i
,
u
1
, u
1
, 0
deg
id − K
i
,
u
3
, u
3
, 0
deg
id − K
i
,B
a
2
,r
, 0
deg
id − K
i
,B
u
1
,r
, 0
1 1
−1
l
−1
.
3.14
So we have −1
l
1. It is impossible. From the above discussion, we conclude that there
must exist another critical point u
∗
1
∈ u
1
, u
3
, which satisfies u
∗
1
/
u
1
and is nonconstant.
Similarly, we can discuss the order interval u
3
, u
5
, and we get another critical point
u
∗
2
/
u
2
. We let the procedure go on.
This completes the proof of Theorem 3.4.
Thus, we prove that the conclusion of Theorem 1.1 holds.
The Proof of Corollaries 1.2 and 1.4.
Proof. See Theorem 3.5 of Li 1.
Proof of Theorem 1.3. From the variational point of view, solutions of p
2
are the critical points
of the functional
J
2
u
1
p
Ω
|
∇u
|
p
dx
α
p
Ω
|
u
|
p
dx
1
p
∂Ω
b
x
|
u
|
p
ds −
Ω
F
x, u
dx,
3.15
defined on X : W
1,p
Ω,whereFx, u
u
0
fx, sds.
We show that J
2
belongs to C
1
X, R.Infact,weset
J
21
u
1
p
Ω
|
∇u
|
p
dx
α
p
Ω
|
u
|
p
dx −
Ω
F
x, u
dx, J
22
u
1
p
∂Ω
b
x
|
u
|
p
ds.
3.16
Under the condition f
1
,itiswellknownthatJ
21
is a C
1
-functional. Next, we consider J
22
.If
we let u, v ∈ X,0< |t| < 1,
J
22
u tv
− J
22
u
t
∂Ω
b
x
|
u
|
p−2
uv ds
q≥2
C
q
p
p
t
q−1
∂Ω
b
x
|
u
|
p−q
|
v
|
q
ds
−→
∂Ω
b
x
|
u
|
p−2
uv ds,
t −→ 0
.
3.17
So we have that J
22
has a Gateaux derivative and J
22
u,v
∂Ω
bx|u|
p−2
uv ds.
Boundary Value Problems 13
Let u
n
→ u in X;now,byH
¨
older’s and Sobolev’s inequalities we can estimate
J
22
u
n
− J
22
u
,v
∂Ω
b
x
|
u
n
|
p−2
u
n
−
|
u
|
p−2
u
vds
≤b
L
∞
∂Ω
∂Ω
|
u
n
|
p−2
u
n
−
|
u
|
p−2
u
v
ds
≤
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
cb
L
∞
∂Ω
∂Ω
|
u
n
|
|
u
|
p−2
|
u
n
− u
||
v
|
ds if p ≥ 2,
cb
L
∞
∂Ω
∂Ω
|
u
n
− u
|
p−1
|
v
|
ds if p<2,
≤
⎧
⎪
⎨
⎪
⎩
cb
L
∞
∂Ω
T
u
n
− u
L
p
∂Ω
Tv
L
p
∂Ω
if p ≥ 2,
cb
L
∞
∂Ω
T
u
n
− u
p−1
L
p
∂Ω
Tv
L
p
∂Ω
if p<2,
3.18
but, when p ≥ 2, p
p/p − 1 ≤ p,wehaveu
L
p
≤u
L
p
,thenwehave
J
22
u
n
− J
22
u
,v
≤
⎧
⎪
⎨
⎪
⎩
cb
L
∞
∂Ω
T
u
n
− u
L
p
∂Ω
Tv
L
p
∂Ω
if p ≥ 2,
cb
L
∞
∂Ω
T
u
n
− u
p−1
L
p
∂Ω
Tv
L
p
∂Ω
if p<2,
≤
⎧
⎪
⎨
⎪
⎩
cb
L
∞
∂Ω
u
n
− u
W
1,p
Ω
v
W
1,p
Ω
if p ≥ 2,
cb
L
∞
∂Ω
u
n
− u
p−1
W
1,p
Ω
v
W
1,p
Ω
if p<2,
3.19
where 1/p 1/p
1, T : W
1,p
Ω → L
p
∂Ω is trace operator, and Tu
L
p
∂Ω
≤ Cu
W
1,p
Ω
for all u ∈ W
1,p
Ω with the constant C depending on Ω by Sobolev Trace Theorem see 6.
To get 3.18, we have used the following well-known inequalities:
|
u
|
p−2
u −
|
v
|
p−2
v
≤
⎧
⎨
⎩
c
|
u
|
|
v
|
p−2
|
u − v
|
if p ≥ 2,
c
|
u − v
|
p−1
if p<2,
3.20
which hold for a convenient c>0, u, v ∈ R
n
.So
J
22
u
n
− J
22
u
≤
⎧
⎨
⎩
cb
L
∞
∂Ω
u
n
− u
W
1,p
Ω
if p ≥ 2,
cb
L
∞
∂Ω
u
n
− u
p−1
W
1,p
Ω
if p<2.
−→ 0,
n −→ ∞
. 3.21
So J
22
u is continuous and J
2
∈ C
1
X, R.
14 Boundary Value Problems
Consider the truncated functions
f
x, t
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
0,t≤−M
2
,
f
x, t
, −M
2
≤ t ≤ M
1
,
0,t≥ M
1
3.22
and the corresponding functional
J
u
1
p
Ω
|
∇u
|
p
dx
α
p
Ω
|
u
|
p
dx
1
p
∂Ω
b
x
|
u
|
p
ds −
Ω
F
x, u
dx,
3.23
Fx, t
t
0
fx, sds.
From Perera 18 we have that
J satisfies PS condition. From the deformation
theorem, we know that
J satisfies deformation property when
J satisfies PS condition. By
a similar discussion as in Theorem 1.1, we only need to discuss the critical points of
J.
Now, we construct the sub-sup solutions of p
2
. It is easy to see that M
1
is a constant
sup-solution of p
2
and −M
2
is a constant subsolution. Moreover, we consider εϕ
1
for all
ε>0 small enough. From 14 we know that ϕ
1
x > 0, x ∈ Ω. In fact, with u : εϕ
1
,byg
1
we have
−Δ
p
u α
u
p−2
u − f
x, u
ε
p−1
ϕ
p−1
1
x
⎡
⎣
λ
1
α
−
f
x, εϕ
1
ε
p−1
ϕ
p−1
1
⎤
⎦
≤ 0. 3.24
Furthermore, ϕ
1
∈ W
1,p
Ω ∩ L
∞
Ω satisfies −Δ
p
ϕ
1
λϕ
p−1
1
in the weak sense, then the
regularity theory for the p-Laplacian e.g., 19 implies ϕ
1
∈ C
1,α
Ω for some α αn, p ∈
0, 1.Moreoverϕ
1
≥ 0. In addition, by the strong maximum principle of 20 and ϕ
1
/
0, then
ϕ
1
> 0inΩ and ∂ϕ
1
/∂ν < 0on∂Ω.Soifbx is small enough at some point x
0
∈ ∂Ω,wecan
have |∇u
|
p−2
∂u/∂νbx|u|
p−2
u|
x
0
≤ 0. From the above discussion, we have a sub-solution
of p
2
, a.e., εϕ
1
x
0
.Withεϕ
1
: εϕ
1
x
0
.
By a similar argument we can find that −M
2
, −εϕ
1
is a pair of strict sub-sup solutions.
Now we study the order interval −M
2
,M
1
in X which includes two suborder
intervals −M
2
, −εϕ
1
and εϕ
1
,M
1
.ByLemma 2.2, there exists weak solutions of p
2
relative minimum points u
2
, u
3
in −M
2
, −εϕ
1
and εϕ
1
,M
1
, respectively. We can infer
that
Ju is bounded from below on −M
2
,M
1
, so we get a mountain pass point u
1
∈
−M
2
,M
1
\−M
2
, −εϕ
1
∪εϕ
1
,M
1
according to mountain pass theorem in order interval.
From the definition of mountain pass point, we have that C
1
J,u
1
is nontrivial.
From assumption g
2
, we know that the left and the right derivatives of
f at 0 are
different, we consider the problem
−Δ
p
u
f
x, u
− α
|
u
|
p−2
u, in Ω,
|
∇u
|
p−2
∂u
∂ν
b
x
|
u
|
p−2
u 0, on ∂Ω,
3.25
Boundary Value Problems 15
where
f ∈ C
Ω × R,andasu → 0wehave
f
x, u
− α
|
u
|
p−2
u
f
x, 0
− α
u
p−1
−
f
−
x, 0
− α
u
−
p−1
◦
|
u
|
p−1
. 3.26
We take a
f
x, 0 − α, b
f
−
x, 0 − α; then also from assumption g
2
and the definition of
Σ
p
, we know that a, b /∈ Σ
p
.
Then we consider the following cases.
1 If a, b does not belong to Σ
p
, but lies above C
2
,then
C
q
J,0
0forq 0, 1 3.27
by Lemma 2.6iv.Inthiscase,C
1
J,00, so C
q
J,0 C
q
J,u
1
,wehaveu
1
/
0.
2 Denote
J
a−b
u
Ω
|
∇u
|
p
−
a − b
u
p
∂Ω
b
x
|
u
|
p
ds, u ∈ X,
3.28
and
J
a−b
is the restriction of J
a−b
to the C
1
manifold
S
u ∈ X :
Ω
|
u
|
p
1
,
3.29
where a
f
x, 0 − α, b
f
−
x, 0 − α as shown above.
From g
2
, we know that a, b does not belong to Σ
p
,if
J
a−b
u >b,a.e.
J
b
a−b
∅,then
C
q
J,0
δ
q0
Z 3.30
by Lemma 2.7.Inthiscase,C
1
J,00, so C
q
J,0 C
q
J,u
1
, and we have u
1
/
0.
Now, we discuss the solutions in −M
2
,M
1
more deeply. We already have four
solutions 0, u
1
, u
2
, u
3
,whereu
1
is the mountain pass point and u
2
, u
3
are the local minimum
points of J
2
. For the minimum points u
2
, u
3
,wehave
C
q
J,u
2
∼
δ
q0
G, C
q
J,u
3
∼
δ
q0
G. 3.31
Since u
1
is a mountain pass point, for the Leray-Schauder degree of id −
K,wehave
the computing formular
deg
id −
K, B
u
1
,r
, 0
−1, 3.32
where r>0 is small enough,
K −Δ
p
m αg
p
·
−1
f
∗
|
X
: X → X is of class C
0
and
16 Boundary Value Problems
strongly order preserving, f
∗
x, ufx, umg
p
usee Remark 2.5b. T hen according
to Poincar
´
e-Hopf formula for C
1
case and the computation of C
q
J,0,wehave
index
J,0
−1
d
l−1
. 3.33
From the additivity of Leray-Schauder degree and Theorem 1.1 in 10,wecanget
1 deg
id −
K,
−M, M
, 0
deg
id −
K,
−M, −εϕ
1
, 0
deg
id −
K,
εϕ
1
,M
, 0
deg
id −
K, B
0,r
, 0
deg
id −
K, B
u
1
,r
, 0
1 1
−1
d
l−1
−1
.
3.34
It is impossible. From the above discussion, we conclude that there must exist another critical
point u
∗
∈ −M
2
,M
1
, which satisfies u
∗
/
u
1
and is nontrivial.
This completes the proof of Theorem 1.3.
Proof of Theorem 1.5. Consider the truncated function
f
i
x, t
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
0,t<0,
f
x, t
, 0 ≤ t ≤ a
i
,
f
x, a
i
,t>a
i
.
3.35
Corresponding functional is
J
i
u
1
p
Ω
|
∇u
|
p
dx
α
p
Ω
|
u
|
p
dx
1
p
∂Ω
b
x
|
u
|
p
ds −
Ω
F
i
x, u
dx,
3.36
where F
i
x, u
u
0
f
i
x, sds, i 1, 2,
It is known that the solution of p
2
is also a solution of the following equation as in
the same discussion in Lemma 3.2:
−Δ
p
u α
|
u
|
p−2
u f
i
x, u
, in Ω,
|
∇u
|
p−2
∂u
∂ν
b
x
|
u
|
p−2
u 0, on ∂Ω.
3.37
Boundary Value Problems 17
By the standard argument w e know that J
i
satisfies J
1
–J
3
and the order intervals consisted
by sub-super-solutions are admissible invariant set of J
i
. Taking v
0
−M
2
, v
1
a
1
> 0, then
J
i
u has a minimizer u
1
∈ v
0
,v
1
. By assumption F
there exists a t
1
> 0suchthat
J
2
t
1
ϕ
1
t
p
1
p
Ω
∇ϕ
1
p
dx
α
p
t
p
1
Ω
ϕ
1
p
dx
t
p
1
p
∂Ω
b
x
ϕ
1
p
ds −
Ω
F
x, t
1
ϕ
1
dx
≤
λ
1
α
t
p
1
p
Ω
ϕ
1
p
dx −
λ
1
α ε
0
C
t
p
1
p
Ω
ϕ
1
p
dx < J
2
u
1
.
3.38
If we take v
2
a
n
1
>t
1
ϕ
1
,wheren
1
<i,then
J
i
t
1
ϕ
1
J
2
t
1
ϕ
1
<J
i
u
1
3.39
which implies that J
i
u has a minimizer u
2
∈ v
0
,v
2
\ v
0
,v
1
such that J
i
u
2
<J
i
u
1
.By
Lemma 2.3 we get a mountain pass point u
3
.Moreover,v
0
<u
i
<v
2
, i 1, 2, 3, and u
i
are
positive.
Next, we take v
1
a
n
1
, v
0
εϕ
1
.ThenJ
i
u has a minimizer u
2
∈ v
0
,v
1
.By
assumption F
there is a t
2
> 0suchthat
J
2
t
2
ϕ
1
<J
2
u
2
. 3.40
If we take v
2
a
n
2
>t
2
ϕ
1
,wheren
2
<i,then
J
i
t
2
ϕ
1
J
2
t
2
ϕ
1
<J
i
u
2
3.41
which implies that J
i
u has a minimizer u
4
∈ v
0
,v
2
\ v
0
,v
1
such that J
i
u
4
<J
i
u
2
.By
Lemma 2.3 we get a mountain pass point u
5
.Moreover,v
0
<u
i
<v
2
, i 1, 2, 3, 4, 5, and u
i
are
all positive. Continue making the procedure we obtain the result.
The proof is complete.
Corollary 3.5. Moreover, p
1
has infinitely many nonconstant negative energy solutions {u
k
},
which are mountain pass types, if the conditions as in Theorem 1.1 hold and J
1
a
2k
→−∞or
J
1
b
2k
→−∞as k → ∞.
Proof. Assume that J
1
a
2k
→−∞as k → ∞.Letc inf
γ∈Γ
max
γI∩S
J
1
ut,whereΓ
{γ ∈ CI, W|γ0a
2k−1
,γ1a
2k1
},andI 0, 1, S W \ W
1
∪ W
2
, W u
2k−1
, u
2k1
,
W
1
u
2k−1
, u
2k−1
, W
2
u
2k1
, u
2k1
, c
∗
Ja
2k
, k 1, 2, Wediscusstheproblem
in W which have two minimum points a
2k−1
and a
2k1
.Wehavethata
2k−1
and a
2k1
are in
the same radial direction A {ke
1
| k ∈ R}, e
1
is the first eigenvalue function of −Δ
p
α
with Neumann boundary. In fact, e
1
is a constant. We conclude that c
∗
≥ c see Corollary 3.4
of C. Li and S. Li 21.Furthermore,iff
3
, f
4
hold, then c
∗
>c.Infact,ifc
∗
c,then
c
∗
max
u∈γ
∗
I∩S
Juinf
γ∈Γ
max
γI∩S
Jut Ja
2k
,whereγ
∗
is a special path between
a
2k−1
and a
2k1
, which is a path of radial direction A {ke
1
| k ∈ R}.Soa
2k
is a mountain
pass point. But according to assumptions f
3
and f
4
, we know that C
1
J
1
,a
2k
0, l
/
2,
that is, a
2k
is not a mountain pass type. This is a contradiction. We draw the conclusion.
18 Boundary Value Problems
Remark 3.6. In Theorems 1.3 and 3.4, we can deal with the case in which a, b lies above C
2
,
but when a, b lies between C
12
and C
2
,then
C
q
J
1i
,a
2
∼
⎧
⎨
⎩
Z, q 1,
0,q
/
1,
∼
C
q
J
1i
,u
1
,
C
q
J,0
∼
⎧
⎨
⎩
Z, q 1,
0,q
/
1,
∼
C
q
J,u
1
,
3.42
we cannot distinguish u
1
from a
2
and 0, then there may not be nonconstant solutions and
nontrivial solutions to p
1
and p
2
.
Remark 3.7. If we give the assumption
F
Ω
Fx, udx > μ
2
ε
0
/2
Ω
u
2
dx,asu≥M, u ∈ E
2
,whereE
2
{u ∈ E |
u k
1
e
1
k
2
e
2
}, e
1
,e
2
are the first and the second eigenfunctions of −Δ
p
α} with
Neumann boundary, respectively, for all k
1
,k
2
∈ R, e
1
e
2
1, ε
0
> 0, and M
is large enough,
then under f
1
–f
5
and
F, we can obtain infinitely many nonconstant positive, negative,
and sign-changing solutions of p
1
.
As a matter of fact, we can infer
F from F.
Acknowledgment
J. Zhang is supported by YJSCX 2008-157 HLJ. X. Xue is supported by NSFC 10971043.
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